The Nature of Algebraic Topology

Both Point Set Topology and Algebraic Topology attempt to describe and
analyze the properties
of geometric objects which are invariant under continuous mappings. Algebraic
Topology does this using an indirect approach. Each geometric object is
associated with a set of algebraic structures, usually groups. Questions
about the geometric objects are converted into questions about the associated
groups. This strategy is analogous to the way transform methods are used to
solve differential equations.

Laplace and Fourier transforms are used to convert a differential equation
into a strictly algebraic equation. Algebraic methods are used to solve for
the transform of the solution to the differential equation. The inverse
transform is applied to the transform of the solution to get the solution to
the differential equation.

In algebraic topology each topological space X has a group (or a sequence
of groups)
associated with it, say G(X) (or {Gn(X); n=0,1,2,..N}). Some information
is lost in moving from the geometric object to the associated group.
An example will be given below to illustrate what all of this means.

Let X be a sphere in 3-space and let Y be a torus also in 3-space.
The groups to be constructed are based upon loops that begin and end
at some point P outside of X and Y. A loop is just a directed path in which
the the beginning point and the end point are the same point. Two loops, x1 and
x2, can be added by attaching the beginning part of the second
loop to the ending part of the first loop. There is also a zero loop,
one that does not leave the point P. The situation is illustrated in the
diagram below.

One loop is equivalent to another loop if it can continuously transformed
into the second loop without intersecting the geometric object. The name for
a continuous transformation is homotopy. In the case
of the sphere any loop may be shrunk into the point P, the zero loop. Thus
all loops are homotopically equivalent to the zero loop.

The group elements are the equivalence classes of the loops. As indicated
above, for the sphere there is only one equivalence class of loops, which
includes the zero loop. Therefore group for the sphere is the trivial group,
the group comprised of only one element [0] and such that [0]+[0]=[0].

For the torus the loops that do not go through the hole of the torus
are all homotopically equivalent to the zero loop. But the loops that go
through the hole of the torus once are not homotopically equivalent to zero.
However all loops that pass through the hole of the torus once in the same
direction are homotopically equivalent to each other. For the torus the
equivalence classes correspond to the number of times the loop passes
through the hole of the torus in a designated direction. Thus the equivalence
class of the loops which pass through the torus hole n times going from
above to below could be represented as [n]. The equivalence class for those passing through the
hole n times in the opposite direction would be represented as [-n].

If a loop in [n] is concatenated with one in [m] the result is a loop
in the class [n+m]. Likewise if a loop in the class [-m] is concatenated
with one in the class [n] the result is a loop in the class [n-m].

From the above it is clear that the group associated with the torus
is the group of integers with the operation of addition. The zero loop
corresponds to 0 and the inverse of an equivalence class [n] is just
[-n].

To sum up:

the first homotopy group for a sphere is the trivial group

the first homotopy group for a torus is the group of integers
under the operation of addition

Clearly the torus is not topologically equivalent to the sphere because
their first homotopy groups are different. In this case the result is
obvious but there are other cases in which it is not easy to tell whether
or not the geometric objects are equivalent. However, if the associated
groups of two geometric objects are equivalent it does not follow that
the objects are topologically equivalent. For example, a hollow cylinder segment
and a Möbius strip have the same first homotopy group; i.e, that of the
integers under addition, but they are not topologically equivalent; the hollow
cylinder is two-sided whereas the Möbius strip is one-sided.

The association
of topological spaces with sequences of groups may be done in more than
one way. The above example concerned homotopy but there is another
important assignment called simplicial homology.
Any particular assignment of group sequences to topological
spaces is called a functor. The assignments are not arbitrary.
The assignment has to be such that if X and Y are topological spaces and
f is a continuous function that maps X into Y (X f→ Y) then
there exists a homomorphic mapping F of Gn(X) into Gn(Y).
A mapping F of groups is homeomorphic if for every x1 and
x2 in G(X), F(x1*x2) = F(x1)+F(x2),
where * is the binary operation in G(X) and + is the binary operation
in G(Y).

Incidence Numbers

Consider a set of vertices (points) labled 0 to p. An m-simplex is
a sequence of (m+1) of these vertices. An m-simplex is said to be positively
oriented if the sequence of labels for its vertices is an even permutation of
<0,1,2,...,p> and a negatively oriented simplex if it is an
odd permutation. Thus an m-simplex is
positively on negatively oriented on the basis of the evenness or
oddness of the number of transpositions required to put the sequence of its
vertices into the natural ordering
of the vertices of which it is made up. The number of vertices in a
simplex is equal to its dimensionality plus one.
A simplex is labeled by its
dimensionality.
A simplex denoted σm
is a simplex of dimension m.

A complex is a set of m-simplexes for all possible values of m.

The incidence number [σmi, σm-1j]
for an m-simplex with an (m-1)-simplex is 0 or
±1 depending upon whether or not the (m-1)-simplex is a face of the
m-simplex according to the rule shown below.

[σmi, σm-1j] = 0
if σm-1j is not a face of σmi,

[σmi, σm-1j] = 1
if σm-1j has a positive orientation in σmi,

[σmi, σm-1j] = -1
if σm-1j has a negative orientation in σmi,

The incidence numbers for (m-1)-simplexes in m-simplexes thus form a
matrix. Likewise there is a matrix of the incidence numbers of
(m-2)-simplexes in (m-1)-simplexes and so on down to 0-simplexes (vertices)
in 1-simplexes (edges). The remarkable fact is that there is a simple,
strong relationship between the incidence matrices at one level and the
the incidence matrices for the next level down.

A Fundamental Lemma of Algebraic Topology

For all simplexes σmi and σm-2j

Σk[σmi, σm-1k][σm-1k, σm-2j] = 0

Polyhedra and Homology

The chain groups of a polyhedron are not very interesting, being isomorphic
to n-tuples of the coefficient group, where n is the number of simplexes of
a particular dimension. The topological content is in the sets of
cycles and boundaries. A cycle is a chain which has an empty boundary. Boundaries
in particular have no boundary so the set of boundaries are a subset of
the set of cycles. The group of boundaries is a subgroup of the group of
cycles. What is more, the group of boundaries is a normal subgroup of the
group of cycles and hence there is a factor group of cycles with respect of
its normal subgroup of boundaries.

The Betti groups of a polyhedron are these factor groups. Enrico Betti (1823-1892) was
an Italian mathematician who combined research in pure mathematics with his
career as a university professor and administrator and later as a politician. In 1871 he
published a treatise in which he attempted to identify the topology of figures with a
set of numbers which were determined by the homotopy groups of the figure. Poincaré
called these numbers the Betti numbers of the figure. It turned out that the Betti numbers
were not adequate to properly characterize the topology of a figure. However the approach
of characterizing the topology of a figure by its groups remained a viable mathematical
pursuit. Groups became analogous to the integers.